AI 中文总结
研究从圆形边界数据向外亚纯延拓中可见极点簇，开发等高线计数指示方法，由正傅里叶系数构建行列式特征，汇总证据到指示场，证明相关确定性结果，是一种簇认证和成像方法而非全极点恢复过程。
AI 中文摘要
我们从圆形边界数据向外亚纯延拓中开发了一种用于可见极点簇的等高线计数指示方法。该方法从由正傅里叶系数构建的行列式特征开始。在纯有限极点模型中，正确的行列式特征分解为一个多项式，其零点是外部极点的倒数。在存在全纯背景、有限采样和噪声的情况下，单个行列式的根是不稳定的，仅用作局部证据。我们将此证据汇总到倒数极点平面上的标量指示场中：在每个采样点，该场记录以该点为中心的小等高线恰好包围一个经验行列式零点的行列式阶数和偏移的分数。所得场起到极点可见性的采样型成像泛函的作用。固定的超水平集给出可见极点簇，而零维持久同调仅用作阈值稳健的后处理步骤。我们证明了将纯极点等高线计数、儒歇稳定性（Rouché stability）、指示场对比度、固定阈值分量恢复和持久性间隙稳定性联系起来的确定性结果。这些结果解释了为什么具有足够留数和间距的孤立极点会产生稳定的高值分量，而弱的、靠近的、边界附近的或噪声主导的极点可能会给出低的、短暂的或合并的分量。该框架是一种簇认证和成像方法，而不是无条件的全极点恢复过程。
英文摘要
We develop a contour-count indicator method for visible pole clusters in outward meromorphic continuation from circular boundary data. The method starts from determinant characteristics built from positive Fourier coefficients. In the pure finite-pole model, the correct determinant characteristic factors into a polynomial whose zeros are the reciprocals of the exterior poles. In the presence of a holomorphic background, finite sampling, and noise, roots of individual determinants are unstable and are used only as local evidence. We aggregate this evidence into a scalar indicator field on the reciprocal pole plane: at each sampling point, the field records the fraction of determinant orders and shifts for which a small contour centered at that point encloses exactly one empirical determinant zero. The resulting field plays the role of a sampling-type imaging functional for pole visibility. Fixed superlevel sets give visible-pole clusters, while zero-dimensional persistent homology is used only as a threshold-robust post-processing step. We prove deterministic results linking pure-pole contour counts, Rouché stability, indicator-field contrast, fixed-threshold component recovery, and persistence-gap stability. These results explain why isolated poles with sufficient residue and separation generate stable high-value components, whereas weak, close, boundary-near, or noise-dominated poles may give low, short-lived, or merged components. The framework is a cluster-certification and imaging method, not an unconditional all-pole recovery procedure.
